Giải tích thực và topo
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| compact/kəmˈpækt/ | chặt chẽ |
| uniform continuity/ˈjuːnɪfɔːm kɒntɪˈnjuːɪti/ | Continuity in which one delta works for every point of the domain. |
| Cauchy sequence/ˈkɔːtʃi ˈsiːkwəns/ | A sequence whose terms eventually stay arbitrarily close to one another. |
Bounded but no maximum
- $f(x) = x$ on $(0, 1)$ is continuous and bounded, yet never reaches its supremum 1. On $[0, 1]$ the same function attains 1.
- GRE analysis tests whether a theorem's hypotheses hold: compact 紧致的 domain, connectedness, completeness, uniformity.
What is the supremum of $\{1 - 1/n : n \ge 1\}$?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The values increase to 1 without reaching it.
Quantifiers decide the definition
- Continuity at each point: for every $\varepsilon$ and every $x$ there is a $\delta$ (which may depend on $x$).
- Uniform continuity 一致连续: one $\delta$ works for all $x$. $1/x$ on $(0,1)$ is continuous but not uniformly continuous.
![f(x) = x on (0, 1) never reaches 1; on [0, 1] it does](/learn-media/gre_mathematics/assets/0501-sup.png?v=1792758417)
If one $\delta$ works for every point, the function is ____ continuous.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The definition of uniform continuity.
Compactness
- In $\mathbb R^n$, compact $\iff$ closed and bounded (Heine–Borel). In a general metric space, closed and bounded is not enough: $[0,1] \cap \mathbb Q$ is closed and bounded in $\mathbb Q$ but not compact.
- The continuous image of a compact set is compact, so a continuous real function on a compact set attains its maximum and minimum.
Compact subsets of ℝ
Apply Heine–Borel to subsets of the real line.
Which subset of $\mathbb R$ is compact?
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It is closed (it contains the limit 0) and bounded.
Every continuous function on $(0, 1)$ is bounded.
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False. (0, 1) is not closed, so the extreme value theorem does not apply: f(x) = 1/x is continuous there but grows without bound as x → 0⁺.
Connectedness and completeness
- Connected: no split into two disjoint nonempty open pieces. Continuous images of connected sets are connected, which gives the intermediate value theorem.
- Complete: every Cauchy sequence 柯西序列 converges in the space. $\mathbb Q$ is not complete. $(0,1)$ is not complete but is homeomorphic to $\mathbb R$, which is: completeness is a metric property, not a topological one.
$x_n = 1/n$ in $(0,1)$ is Cauchy but its limit 0 is outside the space. So $(0,1)$ is not complete and not closed in $\mathbb R$, hence not compact.
Which metric space (usual distance) is complete?
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A closed subset of ℝ is complete.
Common traps
- Applying Heine–Borel outside $\mathbb R^n$.
- Swapping quantifiers: "there is a $\delta$ for all $x$" is uniform continuity, a stronger claim.
Recap
- Compact domain $\Rightarrow$ extrema attained; open intervals give counterexamples.
- Connectedness gives the IVT; completeness is about Cauchy sequences.
- Check every hypothesis before using a conclusion.
- uniform continuity: Continuity in which one delta works for every point of the domain.
- Cauchy sequence: A sequence whose terms eventually stay arbitrarily close to one another.